Question

The condition for which $ {{a}^{2}}{{x}^{4}}+b{{x}^{3}}+c{{x}^{2}}+dx+{{f}^{2}} $ may be a perfect square, is

• A. $ 4{{a}^{2}}c-{{b}^{2}}=8{{a}^{3}}f $
• B. $ 4{{a}^{2}}c=8{{a}^{3}}f $
• C. $ 2{{a}^{3}}c={{a}^{3}}f $
• D. none of these

Answer 1

Answer: (A)

We have, $ {{a}^{2}}{{x}^{4}}+b{{x}^{3}}+c{{x}^{2}}+dx+{{f}^{2}} $
$ ={{(a{{x}^{2}}+cx+f)}^{2}}a $ a perfect square $ ={{a}^{2}}{{x}^{4}}+2ac{{x}^{3}}+(2af+{{c}^{2}}){{x}^{2}}+2cfx+{{f}^{2}} $
$ \therefore $ $ b=2ac,\,\,c=2af+{{c}^{2}},\,\,d=2cf $
and Again $ 4{{a}^{2}}c=4{{a}^{2}}(2af+{{c}^{2}})=8{{a}^{3}}f+{{b}^{2}} $
$ (\because \,\,b=2ac) $
$ \therefore $ $ 4{{a}^{2}}c={{b}^{2}}+8{{a}^{3}}f $